I've recently had a player drop out of a game I'm running. His character is a good character (good story and background, excellent RP hooks, and is partly the party face), and the party would hate to lose him.

One of the other players has volunteered to take over that character (and sacrifice his own). Before I even figure out how I feel about that, is that even allowed on the site? Are player's characters their intellectual property - I know IP was a hot topic recently about the private games, and this seems to have some of the same pitfalls. How about if there is permission?

I would imagine that even if they are intellectual property, copyrighted, or whatever, RPing them wouldn't matter. A loose parallel might be "fanfics" - the authors don't own the rights to their characters, settings, etc, and in many cases butcher them mercilessly, but so long as they don't sell the stories or whatever, it's OK. (This is me having no idea, btw )

But as far as I know that stuff happens from time to time in games (never to me, luckily).

First of all, while the original character concept and backstory would technically fall under IP of the creator, having someone else run that character is no different than the GM running a pre-generated NPC, or players running pre-generated PCs, provided the creator of said character does not ask that the character not be used. As long as the creator approves the use and transferrence, you are covered.

Now, with that said...this is a touchy subject. Most people will not run someone else's character because it isn't theirs. I've done it once, and I never found that I was playing him right. It just wasn't my character.

On top of that, you have a player who has volunteered to play the other character...at the expense of them dropping their own. This tells me that this player is unhappy with their current character and is looking for a reason to play something else. That has all kinds of badness written all over it. You should sit down with this player (figuratively speaking, seeing as it is PBP) and find out why they are so willing to get rid of their own character just to have this other one stick around.

And on that note - have you thought of running the character as an NPC?